[Paper Review] A Fixed Points Approach to stability of the Pexider Equation

Elhoucien Elqorachi, John Michael Rassias+1 more|arXiv (Cornell University)|Jun 13, 2014

Functional Equations Stability Results40 references3 citations

TL;DR

This paper establishes the Hyers-Ulam-Rassias stability of the generalized Pexider functional equation using a fixed point method in complete β-normed spaces. By applying a fixed point theorem with a contractive mapping, it proves the existence and uniqueness of solutions to the generalized Pexider equation under a control function condition, extending stability results to quadratic and Jensen-type equations with explicit error bounds.

ABSTRACT

Using the fixed point theorem we establish the Hyers-Ulam-Rassias stability of the generalized Pexider functional equation $$\frac{1}{\mid K\mid}\sum_{k\in K}f(x+k\cdot y)=g(x)+h(y),\;\;x,y\in E$$ from a normed space $E$ into a complete $β$-normed space $F$, where $K$ is a finite abelian subgroup of the automorphism group of the group $(E,+)$.

Motivation & Objective

• To investigate the Hyers-Ulam-Rassias stability of the generalized Pexider functional equation in vector spaces.
• To extend existing stability results for quadratic and Jensen-type equations using a fixed point approach.
• To establish explicit error estimates for approximate solutions in complete β-normed spaces.
• To generalize prior results on the Pexider equation under a general control function condition.
• To provide a unified framework for stability using fixed point theorems with Lipschitz constants less than one.

Proposed method

• The authors apply a fixed point theorem in a complete generalized metric space with a strictly contractive mapping having Lipschitz constant L < 1.
• They define a generalized metric on the space of functions and use the fixed point alternative to prove convergence to a unique fixed point.
• The method involves constructing a sequence of functions and proving its convergence to a solution of the generalized Pexider equation.
• The control function φ(x,y) bounds the deviation of the approximate solution from the exact equation, satisfying a homogeneity condition involving K and β.
• The solution is decomposed into a quadratic part q(x), a Jensen-type part j(x), and additive components, with error bounds derived via iterative estimation.
• The approach generalizes earlier results by Cădariu and Radu, applying the fixed point method to the Pexider equation with automorphism group actions.

Experimental results

Research questions

• RQ1 Under what conditions does an approximate solution to the generalized Pexider equation on a normed space remain close to an exact solution?
• RQ2 Can the fixed point method be effectively applied to prove Hyers-Ulam-Rassias stability for the generalized Pexider equation?
• RQ3 What are the explicit error bounds between an approximate solution and the true solution in β-normed spaces?
• RQ4 How does the structure of the automorphism group K affect the stability and decomposition of solutions?
• RQ5 Can the method be extended to recover known stability results for quadratic and Jensen equations as special cases?

Key findings

• For a mapping f satisfying the inequality ∥(1/|K|)∑_{k∈K} f(x+k·y) - g(x) - h(y)∥_β ≤ φ(x,y), there exists a unique solution q(x) to the generalized quadratic equation (1.2) and a unique solution j(x) to the generalized Jensen equation (1.3).
• The error between f(x) and the sum q(x) + j(x) + g(0) + h(0) is bounded by (2/2^β)(1/(1−L))χ(x,x) + (1/2^β)(1/(1−L))ψ(x,x), where χ and ψ are defined functions of the control function φ.
• When φ(x,y) ≤ θ(∥x∥^p + ∥y∥^p) with 0 < p < 2β−1 and 1/2 < β < 1, the error bounds become explicit in terms of ∥x∥^p, with constants involving 2^β, 4^β, and powers of 3.
• For the case K = {I, σ}, the solution decomposes into a quadratic part q(x) and a skew-symmetric Jensen part j(x) satisfying j(σ(x)) = −j(x), with explicit error bounds in terms of ∥x∥^p.
• The method recovers known stability results for the additive, quadratic, and Jensen equations as special cases when K is chosen appropriately.
• The fixed point approach yields optimal error estimates under the condition ∑_{k∈K} φ(x+k·x, y+k·y) ≤ (|2K|)^β L φ(x,y), ensuring convergence via contraction.